The Great Equations: Breakthroughs in Science from Pythagoras to Heisenberg

The title of this book is very deceptive. Rather than covering the mathematics in the equations referred to in the title, the author uses them as a springboard for his personal ruminations about life and philosophy. If I were interested in Crease’s “thinking great thoughts,” I would certainly not have bought this book. I expected hard math; I got soft mush. I suppose I should have learned that he is a professor of philosophy before I bought the book and perhaps that is one of the shortcomings of electronic books: if I had been in a bookstore and picked up the physical book I would have checked the book jacket and discovered this fact.

The very first chapter, which uses the Pythagorean Theorem as its launch point, illustrates the character of the book. The author initially tells us that “proof is not an assertion of authority but an acknowledgement of intellectual democracy.” (26) Oh Really! What does that mean? He then advances a loose theory of the raptures one experiences in discovering mathematical proofs. He tells us that there are hundreds of proofs of the Pythagorean Theorem and he even starts to lay out one of them -- the proof used by Euclid in his Elements. But without even attempting to tell the reader how this proof is done, he abruptly abandons the details of Euclid’s proof and starts discussing historical figures like Pascal, Plato, Schopenhauer, Hegel, and ancient Muslim, Chinese, and Indian mathematicians to show how well read and politically correct he is. So by the time one has finished reading the chapter on the Pythagorean Theorem and the following “Interlude,” the reader has not seen the details of a single mathematical proof. All the reader has got is Crease’s personal views of life and its “unforgettable moments.”
In the chapter on Newton’s law of motion, Force = Mass x Acceleration (F = ma), Crease goes on and on about Aristotle and describes the primitive and totally incorrect way in which he saw the physical world. What this has to do with Newton’s brilliant discovery and its proof I could not tell. Crease then goes into raptures over Aristotle:
“What appeared to matter in understanding the motions of nature was the role that things like form, matter, and purpose play in converting potentiality into actuality. And these ultimately referred to the unmoved mover, who communicates through love via the outer spheres to the moon and then to the sublunary world.” (51)

He actually wrote those words! Hello-o-o! this is supposed to be a book about mathematical equations! In fact, I question whether Crease has any but a superficial understanding of the detailed mathematics involved in any of these equations. In his description of Newton’s law of gravitation, for example, he tells of a meeting in which Newton said that the path of the planets around the sun was “an ellipsis.” (79) Is this a typo or an indication that the author really doesn’t know the difference between an ellipse and an ellipsis? Crease later makes a failed effort to explain exponents, a topic so basic that one wonders why he would attempt to explain it at all:
“For instance, 22 = 2 2 2 = 4, 23 = 2 2 2 2 2 = 8, 24 = 2 2 2 2 2 2 2 = 16, and so forth.” (99)
Is this another typo? Or does he really not know that two squared is two times two, and two raised to the power of three is two times two times two?
In the chapter on Euler’s famous equation (℮πi + 1 = 0) the author points out that “there were numerous mathematical advantages” in choosing the constant ℮ as the base of the natural logarithm. Crease then goes on to say that one of those advantages is that the function ℮x can be calculated by use of a complicated infinite series (100). But he fails to say why this is an advantage. It is a totally counterintuitive concept that he leaves unexplained and this just a page after he thought it necessary to explain what 2 squared means! He then writes that the equation (sin x)2 + (cos x)2 = 1 is one of the “more or less obvious properties of sines and cosines.” (101) -- obvious to the same reader who had to have 2 squared explained a page earlier? I suppose that it may be “obvious” to one who is familiar with the unit circle on Cartesian coordinates, that the radius with a value of 1 is the hypotenuse of a right triangle, and that the other two sides, when divided by 1, thus become the sine and cosine values of the acute angle formed by the radius and the x-axis.

The author does go through some of the mathematical gyrations used by Euler in developing his famous equation, but Crease’s descriptions are wooden and empty of any real insight. The essence of it appears to be that Euler translated “many mathematical terms and expressions into the language of infinite series.” (98). He discovered that the trigonometric functions, certain constants like π and ℮, as well as ℮ raised to powers, all can be expressed in infinite series. So he raised ℮ to the power of x times i and, low and behold, he found the sine and cosine series embedded in this expression, and by simple factoring and substitution concluded that ℮xi = cos x + i sin x. Euler then substituted π for the x and the result was his famous equation. Okay, so what? Crease is supposed to answer that question. Why is this equation important or significant? It is not enough for him to point out that it “establishes the deep connection between the exponential and trigonometric functions.” (102) That statement doesn’t mean anything to anyone other than mathematicians, who hardly need Crease to explain things to them. Perhaps the most that he can say about the Euler equation is that it is “simple, elegant, and pleasing.” (93)

Another troublesome part of the book is when the author attempts to explain the sine function. On page 100 he takes us through 90 degrees, then 180 degrees, and so forth on the unit circle, completing the period of the sine wave at 360 degrees. A few pages later (103) he writes: “The sine of π is 0, and the cosine of π is –1.” Well, Mr. Crease, π has a particular value, which is approximately 3.14159. The sine of 3.14159 degrees is certainly not zero. It happens that a circle may be measured in units other than degrees and one of these units is the radian. It also happens that there are 2π (6.2831) radians in a circle. The sine of π radians, the equivalent of 180 degrees, is zero, but Crease does not tell his readers that. One wonders if he even knows that or just read in a book somewhere that the sine of π is zero without any understanding that this statement refers to π radians.
In the gravitation chapter and its Interlude the author makes no reference to any of the math involved in gravitation or how Newton developed it, and the reason may be that he hasn’t a clue as to what the math is. The same is true for the chapter on thermodynamics, which is a truly awful exposition of nothing. This chapter is a classic example of an author failing to understand what it is that his readers don’t understand. It is a blizzard of meaningless words that he apparently extracted from the literature on thermodynamics and put together in the context of little biographies of the principal players in what he calls a “Shakespearian” drama. And of course the only equation in the chapter is the general statement he uses to introduce the topic – S’ – S ≥ 0 – which gets no further attention from Crease. He says, however, that if you do not understand the second law of thermodynamics you are an appalling ignoramus:
“This law is essential to the activities of the world around us. If you do not understand this, you can have little understanding of how the world works.” (111)
I daresay that if you walked up to people on the street of a large sophisticated city not one in a thousand would have a clue as to what the second law of thermodynamics is. The author himself has little better understanding. Another puzzling shortcoming in this chapter is its omission of any reference to Joseph Fourier, a French mathematician/physicist who did pioneering work in heat transfer and wave analysis during the same period covered by Crease. Fourier developed what is generally known today as Fourier analysis, a procedure that involves trigonometric functions and Euler’s constant ℮ and is still in use today.

The entire book reads as a work of personal social commentary, not mathematics. Well, one might ask, should a person be criticized for publishing what he thinks about the social and cultural consequences of mathematical developments without himself being an expert in the fine details of the math? The answer is in general no, with two caveats. First, one should not, in my opinion, write about a difficult complex topic, an area of specialized knowledge, without at least some expertise in the area. Am I qualified to write a book about neurosurgery without any medical education and no experience in the discipline? Sure, with a little research I could put words together in a way to make them sound deep, profound and charged with meaning. But would they add anything to the knowledge base of that topic? And second, if you believe that despite this qualification there should be no constraints on one’s publishing a book on any topic whatsoever, shouldn’t an author so inclined at least familiarize himself with the subject matter before holding himself out as an expert in the field by authoring a book titled “The Great Equations”? Doesn’t a book with this title, written by a university professor, suggest that the author has some specialized knowledge of the mathematics in these equations?

I have read many books on mathematics intended for the non-professional lay person. Authors such as William Dunham, Martin Gilbert, Eli Maor, Marcus du Sautoy, Keith Devlin, Joseph Mazur, and Calvin Clausen, to name just a few, have brought the esoteric world of mathematics to millions of lay readers like myself. But they have done so only because they have a deep understanding of the math and a wonderful facility for explaining it to others. But Mr. Crease has little or no understanding of the math that is central to the equations that he builds his book around. That is a critical failure.

It will take some science background to understand what's here but the stories of the development of the equations are worth reading. I have a graduate degree in science that is too old and unused to appreciate the concepts but the narrative is well worth it.

I am non mathematical, can barely do algebra despite almost 80 years of trying. So I read books on Einstein, the bomb and explanations of deep apace such as hawkin's and Feynman's books for dummies. This book is in the same category, clearly wtitten, excellent explanations going from 1+1=2 to Einsteing's relativity, all the while maintaing one's interest. Enjoyable and recommendable, it is on my Holiday gift list for some of my relatives and not so dummy friends.

Interesting look at things we might,
but shouldn't take for granted that we know.

Easily one of the most interesting books on Mathematica I've ever read. Accessible yet highly informative and entertaining. If you are a Math buff, you will not be disappointed.

It was an excellent book. It fed my new found interest in mathematics as well as provided the history behind the equations. It is something I will reread multiple times.

A fun overview of a handful of important mathematical equations. While the author doesn't go into much depth about the meaning of each equation, he does go into detail about the impact of the equation and why it was important. Any book on this type of subject is going to miss equations that some would believe are important while including some that everyone may not agree should be included; however, Crease does a satisfactory job of justifying why each equation was included. Recommended for anyone who has enough interest in mathematics and physical science to be curious, but those with strong math or science backgrounds can skip this read, as it covers no new ground.

It's a good exercise in logic and a reminder that real flesh and blood, emotional, arrogant, stubborn human beings came jup with these ideas. There's great drama, and I pretty much understood the fundamentals of the primary equations discussed. Many of the equations in the body of the text I (a non-scientific type) just skipped over.